Practical Aspects of Cantilever Retaining Wall Design JOHN W. STILES, Bureau of Public Roads, Federal Highway Administration,

U. S. Department of Transportation

The Bureau of Public Roads publication, "Typical Plans for Retaining Walls" (1), contains a new and simple approach to the relationships between footing dimensions and bearing pressures. A series of graphs show how changes in toe and heel length affect the bearing pressures. These graphs, their construction, and their use are discussed. The graphs, when used in con­junction with the wall dimensions in the typical plans, give the designer a "feel" of how bearing pressure varies with changes in footing width without his becoming involved in computations. Several loading cases are given and the problems associated with each are discussed.

•THE PROCEDURE used in developing the Bureau of Public Roads "Typical Plans for Retaining Walls" (!)is described in this paper. Copies of these "tY.Pical plans" may be obtained from the Superintendent of Documents, U.S. Government Printing Office.

The BPR typical plans present retaining wall dimensions for wall heights from 6 to 30 ft. The first type of retaining wall shown is the cantilever retaining wall on a spread footing. Also shown are pile-supported cantilever walls and counterfort walls. For our discussion we will refer only to the cantilever retaining walls on spread footings. Four loading conditions are shown: case I, level backfill, no surcharge; case ll, level backfill with a 2-ft live load surcharge; case m, backfill on a 2:1 slope, and case IV, backfill on a 1 '/a :1 slope. For the various design dimensions presented, the maximum bearing pressure is also indicated, as shown in Figure 1. (Note that Figure 1 is not the complete sheet from the typical plans but i s only that portion pertaining to this paper. Bar lists, r einforcement layout, and other details have been deleted for clarity.) Be­cause the bearing pressure for a given situation may be intolerable, graphs were in­cluded in the BPR plans. These graphs allow the designer a greater "feel" of how the bearing pressures will vary with changes ~ the width of the retaining wall footing. It is our purpose to explain how to use these graphs, as well as to describe how they were constructed. Appexdix C to this paper includes the notations used.

The graphs, as shown, should only be used in conjunction with the dimensions given in the BPR plans. Any change in the dimensions given, especially in the placement of the stem on the footing, will give erroneous answers. To illustrate the use of the graphs, assume that we have a 30-ft cantilever retaining wall supporting a horizontal backfill. At the site, the maximum allowable bearing pressure is 4 kips per sq ft. In Figure 1 we see that, for the dimensions shown, the maximum bearing pressure is 4. 65 kips per sq ft. Because this exceeds the allowable bearing pressure, we must change the footing dimensions so that the allowable pressure is not exceeded. Entering the graph with a bearing pressure of 4 kips per sq ft and proceeding upward until we reach the H = 30 ft curve, then going horizontally, we see that an increase in toe length of 11 in. will satisfy the requirements. Adding this to the footing-width in the table of wall dimensions, we obtain a new footing width of 17 ft 9 in. If the bearing pressure at the heel is needed, it can be obtained by proceeding downward from the point where the

Paper sponsored by Committee on Substructures, Retaining Walls and Foundations. 87

88

WALL DIMENSIONS

H 8 T D

6 z'- 1o" o·~ .- (:_f 7 J-6 l • O J -.J

d 4-0 1 - 0 t· .J --9 ~ .. :--.-- 1~~

10 , • ., 1 - .J 1 -J

If J-H 1- 6 1 ~ "6

-~ 6~~ ;:~ :~+ 14 7-7 , . , I-~

I~ d-4 ~-0 ,. ,

16 ~ -IQ 2 - 0 1• 6

_!_? . g -1 Z-.!1 r•(;

... '!!___ 10 - 1 ~- J ......!_·± 19 10-~ Z -6 l "' '.I -----r---zo 11 -2 Z-6 r!-=!-~'- 11 - 10 Z--2_ _ Z- · O

!..!.__ ..!!:.. 3 Z-9 z-o !.,_!__ JZ- JO z-g ~ r O

Z4 1.I-5 J-O ~ -3 1---- -N IJ •ll J•ti1 C"--'

~ 14-6 ~"- ~ r--~ 15:!__~:!.:i!..._ ~ ~ 'J.§..:. 9 3 - 6 ___!.:,!_ ~o 16-3 J -6 ~ - "'

30 /6•/0 3-9 3 •9

6 • • lJb4ibf'i(i PRESSURE- k if"• ,,.r .,. (I

QUAJITITJES """ ian CH -~l r;~,-~ -6;ul C• )>{ ••1 b

' MS 16.

OAOI ,. 3

0 4 70 ee_o 0553 , . 7 o 6Z6 ~·. 07-'0 JU 0 d30 3"2 v.~cc 4r1. ~

I 005 560

1104 ... -. I / 90 79. I Z9Z . ., 1 . .Ja1 IU , 7

1. $60 JJZ,6

1. 657 H6.J I 9Sta , .. .J

Z,111 ,,.~ 7

i!.Z39 191. J

Z471 Zl4.7

L"°I ~OiJ

lasz "48.2 Z'99 l.1~ 7

.JZ77 -""'l_ 3<fl5 J.M_IJ . 3. 114 .J'U-?

BEARING PRESSl.R.S FOO OiANGES IN FOOTING WIDTH

EM!!!U ''""' ' ~' Jolul1on

......... ......... H .....

II ,09 ·-,,, 7_ 1. 37 8 1'9 • ,., 10 J.70 II ,., ,. "" "-2.ZI eL U < ,, e,.,.,

'Ii-UJO " '73 10

u• ..!!':. ..!.O~ •O

315 l..tL ~o ...!!..

36' "'-.J.'fl Z4 ..t.t>;'\ .. 4.0.J ....lt.. ~~ -'J.. ~.y. J 4.. ~6 _ ... ••• 30

.JO' Cant,.,""" •~ W6' foJ.i rw""> .,111111r4.,y) flhYJ, "~10·· 11·.17~9· lltµ1,,.,,,. .,t~ ~ (I>}, /hi ,,.,..__ H~I ,,,..,u-. •1 111(/#1 ,,,....._ .1 1.,.~ ,. n

Figure 1. Typical plans for retaining walls.

B

Figure 2. Cross section of case I wall.

horizontal line crosses the curve for heel pressure; in this case the pressure is 1. 77 kips per sq ft. From this informa­tion, the steel requirement for the footing can be modified to meet actual conditions.

80TT_. ASSUMPTIONS

The Rankine method was used in de­termining the soil .fore e acting on the walls. In all cases a well-graded , free -draining backfill material is specified in the plans. This is important because a plastic ma­terial could increase the soil force up to 300 percent of the assumed value. Also necessary are weep holes or another method of draining the backfill. The back­fill material was asswned to have an angle of internal friction equal to 33 deg 41 min, and a unit weight of 120 lb per cu ft.

Using these assumptions, the value of Rankine's constant, £, was determined. The respective values of the soil force, P, were thus determined. In the cases of an inclined backfill, both the horizontal and vertical components of the soil force were used in the analysis. The soil in front of and covering the toe of the wall was neglected becaURP. in some cases it would be nonexistent. This would, in most cases , give an added factor of safety against sliding. If the foundation is a substantial distance below the ground surface, this wedge of soil should be considered.

DEVELOPMENT OF CASE I CURVES

In developing the curves in the typical plans, a series of relationships were de­rived from the wall geometry and loadin con i ons. cross-sectional view of a case I wall and loading is shown in Figure 2.

The generalized equations were derived in terms of known dimensions Hand Q, the soil force, P, and the assigned dimensions D, T, and b. By summing moments of the weightof theareas about the toe ofthefoot­ing (Fig. 3 ), we obtain the relationships given in Table 1.

When all of the weights are summed, the expression will take the form of I;w = YB - M, where Y and M will be constants for this particular wall. After multiplying the weights by the appropriate moment arm, a', andsumming, weobtainanexpres­sion in the form of l;Wa

/ = NB2 - R. It is

noted that, between the weight summation and the moment summation, all terms

Figure 3. Case I weights and moment arms.

Area No .

2

3

4

5

89

TABLE 1

CASE I WEIGHTS AND MOMENT ARMS

Weight Arm

0.15(H - D) T + 0.5

0.00625(H - D)'b (H - D) T+1+-"36b

0.12(Q - D)X x B - 2

0.005(Q - D)2 b T+ 1+~+(\5°)b 0.15BD 0.5B

containing Bare multiplied by B/2; therefore, Y = 2N. This will be necessary in sim­plifying the general equation in the derivation.

By summing forces and moments, "DW/A ±Mc/I, about the center of the footing, we obtain

r.w ± 6 I:W (a - B/2) ± 2PQ - 0 B B2 B2 -

where a is the distance from the toe of the footing to the centroid of the total mass, M = twa, c = B/2, and I= B3/12.

For the condition of zero heel pressure,

F _ 'EW 6 I:W (a - B/2) _ 2PQ _ O h - B + B2 Bz -

Because 'EW is in the form of YB - M and 'EWa / is in the form of NB2

- R, we obtain

Fh = YB2 - MB + 6NB2

- 6R - 3YB2 + 3MB - 2PQ = 0

Substituting Y = 2N and solving for a B that will give a heel pressure of zero, we obtain

By going back to the general equation for any pressure under the footing and applying the correct signs, we obtain a toe pressure equal to

Because the footing width, B, found here is fo1· a heel pressure of zero, let us now see what happens when the heel is increased by a distance, S, added to the heel. The addi­tional weight of concrete and soil to be considered is equal to K = 0.15SD + 0.12(Q - D )S, where the unit weight of soil = 120 pcf, and K = 0.12Q + 0. 030 for each 1-ft increment added. Referring to Figure 2 and again summing moments about the midpoint of the new footing width, we obtain

4 (IiW +KS) 6 r.wa' + 6KSB + 3KS2 - 2PQ

Ft= (B + S) - (B + S):i

90

F __ 2(I;W+KS) 6IIWa'+6KSB + 3KS2-2PQ h - (B + s) + (B + sP

The B found above, for Fh = O, is less than the width required to prevent sliding. Be­cause an increase in the heel length will affect the sliding characteristics more than an increase in the toe length, the heel should be increased up to the point where sliding is impending.

By using a coefficient of friction against sliding of concrete on undisturbed soil of 0.45, we obtain the expression P = 0.45 IIW for the impending of sliding. By applying a factor of safety of 1. 5 against sliding, we obtain a P ~ 0.30 r;,w. Any B that gives a value of r;w less than 3.33P will have to have a key to prevent sl.iding. For the case I walls shown in the typical plans, the value of B given is one that will give r.w "" 3.33P; no key is required.

Consideration should now be given to the pressure under the toe. Any increase in the toe length will reduce this pressure mo1·e substantially than a further increase in the heel length. For these reasons let us now see what effect an increase in the toe length will have on the soil pressw·e .

Summing moments about the midpoint of the footing width at which sliding is impend­ing, plus any increase in the toe length, we obtain

4 (IIW + 0.15SD) Ft = (B + S)

6 r.w (a+ S) + 0.45s2D - 2PQ (B + S) 1

Fii __ 2 (IIW + 0.15SD) 6 IIW (a + S) + 0.45S2D - 2PQ - (B + S) + (B + S) 2

By setting the limit of toe pressure equal to heel pressure, with the increases in toe length we obtain a maximum

s _ B r.w - 2 r;wa' + o. 67PQ - IIW - 0.15BD

SAMPLE CASE I SOLUTION

A sample problem is solved in Figure 4. The solution is as follows:

1. Determine the dimensions H and Q. Assume a footing thickness D (approximately 1/10 H, 1 ft 3 in. minimum). and distance T (apprnxia:iately 6 in, 1 H/lG).

2. Solve the footing width at Fh = 0. In this example B = 9 ft 2 in. 3. Increase the heel length up to the footing width at which sliding is impending (B

in the plans) and solve for the corresponding lH:iaring pressures. The width at which sliding is impending is equal to 11 ft 2 in.

4. Using the footing width at which sliding is impending, 11 ft 2 in., increase the toe length up to the point at which the bearing pressure is uniform.

5. Plot the bearing pressures versus footing width for the various increments of in­creases in footing width, thus obtaining the curves shown in Figure 5.

A sample worksheet is shown in Appendix A for each of the other three cases.

DEVELOPMENT OF CASES II, III, AND IV CURVES

The case of the horizontal backfill is by far the simplest loading on a retaining wall. As surcharges of different shapes and magnitudes are added, the soil force becomes more complicated. If a umtorm surcharge is assumed, as in case II, the soil force is increased by an amount P 1 = yh8 Qt". This force, as shown in Figure 6, is applied at half the height of the wall. By considering this loading condition and reducing it to an equivalent loading system applied at one-third the height, we can use the case I equa­tions by substituting P' = P + P 1•

91

CASE I

tt· H • -2Q:._ Equivalent Fluid Pressure • 0.035 *•' Boller

D • -ilL...... b•i°J, for (H-D) >20'

T • .....u:_ b•f 7, for (H-0)~20' Q. -1.2£_

P • E.F.P. i2 • 6.65~ If P < 0 .3(1W) • 6.653 9 1'cr 8 ·11'i!. ..

X • B-T-1-b(~) • 8 - 4.2604 F.S. again•! sliding> 1.5

I I! I If P<~·= 7419/ r., 8= 9c ;:, ··

"' a' Wa' F.S. againsl overlurning > 2.0

.15(H-D)• 2 . 7:}_75 T·0.5 • .lO 8 .2125 u. 1:i:,~: ~?l

.00625(H-0)2

b• i. 040 8 T"'l•tlfj)b • 3. 7535 3. 9066

.12(Q-O)X • 2.138-9. 0 75 8 - ~ • 0.58 • .21302 l.0658'-19.331 V• !6m2'•§Y~l!-ZPr~r~2 l

.005(Q-0)2b. 0. 7877 18 •$)

T •t•fji(~)b•~ 3./617 A• 1:i:w + Q. 15~QI

.15 BO • Q:i.§:.U. '1. .58. Q ~.e 0.1312 B' IB+Sl 1 W• YB· M • 2.398 - i 5087 -:i.Wo • NB~R · /v1.M· - 4. r1o;;r1/

!6~W!•·~l + Q.45os2-IPOI C•

J~·¥·c•i2-~· (B•S)I!

B • )108. 4 73 + JO.IS 7 + 3.551 - 18845 • 9. 169 fl . 1.1 •• B • 9 C2 "

F, • ~- ~+ ~ ' U.?03-6893.5+3. o.331 = B B - --- -- 3. TS99kipsltt2

Effect on soil pressure when B is increased by incremen1 S io ho o t In IOI!

F1 • 4U-V !;' u v F F• F1 •4A - C s A c 1: F• Fh • V-2U 0 3.16 0

Fh •C-2A I L846 S.034 2 .350 l :U I 1 9165 44290 J.36 0 .54 2 1.726 S. 079 1.82 1.63

B. 9 '-2" 1.99 1.985"9 4.8938 305 0 . 92 B• 11~2" 2296 l 69i 5.080 1.10 1.70

Figure 4. Sample solution for minimum width footing for case I retaining wall.

- • 0 : s z -.. ,!. ~ :

at eel

4 3 2 BEARING PRESSURE - klp1 ptr 1~ II .

Figure 5. Bearing pressures for changes in footing width for case I retaining wall.

92

Uniform Surcharge

Figure 6. Cross section of case 11 loading.

In the first two cases considered, a relatively simple relationship for the mini­mum base width can be obtained. For the case of a sloping backfill, the solution is not so simple. The basic equations for the sloping backfill must consider the sloped soil force. Boih ihe horizontal and vertical components of this force were considered in our analysis. Even though some design offices do not consider the vertical force, we felt that a more realistic condition of loading was ob­tained when this force was considered.

The basic P/ A ±Mc/I equations, for this condition, were derived and, after seeing that the solution would be compli­cated , a minimum base wid th was not solved for. Instead, a value for the base

width was assumed and the values for heel and toe pressure were computed. The equa-tion for the bearing pressures, for any backslope, are

Ft Psine + "iJW

B 6 r.w (a - B/ 2)

B2 3P sine 2P cos 0h

B + B2

P s in 0 - 'f,W 6 tW (a - B/2) 3P s ine 2P cos eh B + B2 + B - B2

For an increa se, S, in the length of the toe, with a 2 :1 slope, the bearing pressures are

4"£W + 0.15DS - 0.8944P 1.7888Ph - 6tWa' - 6sr.w + 0.45DBS Ft = (B + S) + (B + 8) 2

1.7888P - 2tw + 0.15DS 6tWa' + 6SEW - 0.45DBS - 1.7888Ph (B + S) + (B + 8) 2

a nd with a 1 % :1 slope , bearing pressures are

Ft 4"£,W + 0.15DS - 1.1094P 1.6641Ph - 6tWa' - 6Stw + 0.45DBS

(B + S) + (B + S) 2

2.21BBP - 2tw + 0.15DS 6Ewa' + 6SEW - 0.45DBS - 1.664Ph (B + S) + {B + S) 2

Although these equations appear rather complicated, they can be set up in worksheet form, as shown in Appendix A, and solved without too much effort for values of S.

The equations for an increase in the heel. length are unwieldy and it is much easier to go back to the basic equations and compute the soil pressures for the increased heel length.

The worksheets and resulting gr aphs for cases II, III, and IV are included in Appen­dixes A and B respectively. The actual derivation will not be presented because space does not permit and the individual designers may want to change the basic assumptions.

As stated previously, the assumptions may easily be changed for any conditions and similar curves constructed to facilitate future designs.

SUMMARY

93

It was found that a curve expressing bearing pressures, for increases or decreases in footing width, can be constructed for any particular loading condition or wall geom -etry. The relationships between changes in footing width and resulting bearing pres­sures can easily be derived and curves constructed. Although only a limited number of these curves are given, similar curves may be drawn for any particular design criteria. These curves are of particular value when used with a set of standards.

It was noted that in a great number of cases the sliding characteristics were of major concern. For these cases an increase in the heel length is more efficient than an in­crease in the toe length, but for sloping backfills a key is the best solution. A backfill on a steep slope will cause very high soil pressures and more sliding problems than flatter backslopes. For high soil pressures , increases in the toe length will be more effective in reducing soil pressures than increases in heel length.

ACKNOWLEDGMENTS

It is a pleasure to acknowledge the help received from Stanley Gordon and other mem­bers of the Bridge Design Section of the Washington, D. C., office. Without their as­sistance, these charts, and this paper, would not have been possible.

REFERENCES

1. U. S. Bureau of Public Roads. Typical Plans for Retaining Walls. U. S. Govl Print­ing Office, Sept. 1967.

2. Terzaghi, Karl, and Peck, Ralph B. Soil Mechanics in Engineering Practice. Wiley and Sons, 1967.

3. Huntington, W. C. Earth Pressures and Retaining Walls. Wiley and Sons, 1957.

1<

1o1 , .

-

\1•z• 0

clc I 8

w 15(H-O) '

Appendix A

WORKSHEETS FOR CASES II, III, AND IV

~S~rctt4tQt H• --- Equivol1nt fluid pnu = ___ O• ___ P1 • 2Q E-~~ • ---

T • --- p'" P +I !5P1

= ---

liJ Q• _ __

t P• C F,P, f • ---X• B - T - 1-bi~)• ___

I ' Wo'

T+ 0 .5 = II P+P1 s

Bllltr lt= l/4 '"!, lor1H - D)>20'

b• ;. , , lor (H·O)S 20'

0. 3(2W)

00625IH-Dfb, --- T+l+(~)b •---

•+ § ~s aga ln1t 1lidin9 > I 5

.12(Q-O)X' II P's 1.11wo'

,005(Q-Ol2

b '--- r .. 1+-fl+l~lb =-- F.S. 09oin1I ourlurnina > 2 .0 . le BO• .5 e • l: w • YS- M• ··w·-- I. wa· • N&

1- R •

B' ji.p . ,,. ,,r _. II -- UH B• __ Tot PrtHure = ~ -~·+!ff = --- kip/ft

2

Effect on 1oi I '"'ts urt when B ;, increaud by incramtnl s

lo hul to IOI

r, ~(l:W+YS)-11!

11a (6l:wo' + 6YSB - 2P'Q + 3y5

1) r, • 18!51 1Iw+o 15SDJ - 18!512 ~Iwlo+Sl+ 0 .45os" -zp'QJ

Fti :-~(I.W4-YS) + (B~S)2 ( 6 IWa' + 6YSB - 2P'Q + '3YS2

) r, ·- if.s-11:i:w+o 1sso1+ <f.;1• l!;i:wto•Sl~ o 4~os1 -2p'QI

Figure A 1. Solution for minimum width of footing for case 11 retaining walls.

94

Jf Ps~'

"'- - - -o • _ _ _ _

T • - ---u • ___ _

B•-- - ­p : Q.0249h2• ----

J • T+l+i~)b • ---­

L • T+l+ft • --- -

K • Q- 0.5L • - ---

h • K+ 0 .58• ----

Pain& .. _ _ _ _

Pco18• ___ _

T+0 .5 : _____ _

T+l•l.rn'1lb •----­

U~)b • - --- -~3IL+ 28). ____ _

F.S 09oin1t ovutwnino ;t; 2 O

15(H-O) • - ----

00625 blH-0)2

'--­

OOSb(O-Of • - -­

.0318t• L'> - .06BL '--­

,i2\0- Dll5-.ii "'---­.1580 •

11z(B+J) =----­.58 •

2 W•----- ··w·------r • •

Ef l etl on soil pressure when 8 i' increased by mcrement S lo tor

1 Tlll P!t - H~~:s?'W 1 0 .4SpU F • !ZllP-U:W+OISDS + 111 Cl+I)

Bolter

b • \-.111 '~. for (H-0)> 20' b • 1tz 'r. for IH-D)s20'

1in e • 0.4472

COi 9 • 0 .8144

~ lWo •-- ---

Figure A2. Solution for soil pressures for case 111 retaining walls.

A I

F.S 09oin11 overturning~ 2 0

" ' ----0 • ___ _

T • - - --0•----8 • ___ _

P=0.0499h2

• ----

w .16CH-O) •-- ---0062~bllt-oJ" •---­.ooSb(o-oi' • - - -­.04102· 1!> - ,099L •--­

,121Q•OX8- JJ •---­,15 80 a

J • T+l+(~)b: ---­

L • T+l+'ff- • ---­

K•O-iL•---­h • K+fB·-----P1 in8 s ___ _

Pco18• ___ _

.· T+0 . 5 =-----­T•l•l.rn'1lb •---­

L+l~lb • -----1t3ll+ 28) · - - --­

l1z(B+J) =----­.58 •

··w··--- ---

E""' on toil prn1urt when 8 i1 incrtaHd by increment S to tot

4Q. A !IOI - ' 1014 1" + IJ I UP" - tJwf-lll' t 0 .43DH Fi• (l• fl ll <t l

Boller b•lft'~ fOf IH-0)>20' b•'t4>0 for (H-O)s20'

l ine .. o.~547

COi 8 • 0 . 8321

l Wo•--- --

Figure A3. Solution for soil pressures for case IV retaining walls.

95

Appendix B

GRAPHS FOR CASES II, III, AND IV

1

. i . .. ' ;-)( "'

• 3 BEARING PRESSURE - klp1 por ,,. II.

BEARING PRESSURES FOR CHANGES IN FOOTING WIDTH

Figure 81. Case II.

3

• {!.

2 - : •o ~! I u

>- .5

BEARING PRESSURE - kips per sq. ft.

Figure 82. Case 111.

96

{:-( Iv! s• I

BEARING PRESSURE - kipa per aq It.

BEARING PRESSURES FOR INCREASES IN FOOTING WIDTH

B = D =

Fh = Ft = H = J =

L = P-

P1 = Q= s T w

EW X=

M, Y,N,R = a= (

a = b = h = ¢ = y =

,E

a =

footing width depth of footing soil pressure at heel soil pressure at toe

r'igure 83. Case IV.

Appendix C

NOTATION

total height of retaining wall distance from toe of footing to intersection of inside face of wall and the foo ting horizontal distance from edge of toe to point where backfill touch soil fo1 ce ca c i

soil force caused by uniform surcharge height of backfill at face of wall increase in footing l ength distance from edge of toe to face of wall weight of any section of wall or soil total vertical force distance from back of footing to intersection of inside face of the wall and the footing constants for any particular wall geometry distance from toe to centroid of resultant vertical forces distance from toe to centroid of any vertical force batter of inside face of wall vertical height from heel of footing to backfill surface angle of internal friction= 33 deg 41 min unit weight of soil = 120 pounds per cubic foot Rankine 's constant angle backfill makes with horizontal